{"paper":{"title":"Weighted proper orientations of trees and graphs of bounded treewidth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"cs.DS","authors_text":"Ana Silva, Cl\\'audia Linhares Sales, Ignasi Sau, J\\'ulio Ara\\'ujo","submitted_at":"2018-04-11T09:16:44Z","abstract_excerpt":"Given a simple graph $G$, a weight function $w:E(G)\\rightarrow \\mathbb{N} \\setminus \\{0\\}$, and an orientation $D$ of $G$, we define $\\mu^-(D) = \\max_{v \\in V(G)} w_D^-(v)$, where $w^-_D(v) = \\sum_{u\\in N_D^{-}(v)}w(uv)$. We say that $D$ is a weighted proper orientation of $G$ if $w^-_D(u) \\neq w^-_D(v)$ whenever $u$ and $v$ are adjacent. We introduce the parameter weighted proper orientation number of $G$, denoted by $\\overrightarrow{\\chi}(G,w)$, which is the minimum, over all weighted proper orientations $D$ of $G$, of $\\mu^-(D)$. When all the weights are equal to 1, this parameter is equal "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.03884","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}